一般形式可分NMF问题预条件逐次投影算法的鲁棒性分析
Robustness Analysis of Preconditioned Successive Projection Algorithm for General Form of Separable NMF Problem
摘要：
连续投影算法（SPA）在解决可分离非负矩阵分解（NMF）问题中有着良好的应用前景，如文档主题提取和高光谱图像端元检测等。原因之一是该算法对噪声具有鲁棒性。Gillis和Vavasis在[SIAM J. Optim.，25（1），pp. 677-698，2015]预处理器可以进一步增强其噪声鲁棒性。证明的前提是可分NMF问题的维数$d$与分解秩$r$一致.然而，这可能是不现实的期望，该条件成立于可分离的NMF问题出现在实际应用中，在这样的问题，$d$通常大于$r$。本文表明，没有条件$d=r$，预处理的SPA对噪声是鲁棒的。
英文摘要：
The successive projection algorithm (SPA) has been known to work well for separable nonnegative matrix factorization (NMF) problems arising in applications, such as topic extraction from documents and endmember detection in hyperspectral images. One of the reasons is in that the algorithm is robust to noise. Gillis and Vavasis showed in [SIAM J. Optim., 25(1), pp. 677-698, 2015] that a preconditioner can further enhance its noise robustness. The proof rested on the condition that the dimension $d$ and factorization rank $r$ in the separable NMF problem coincide with each other. However, it may be unrealistic to expect that the condition holds in separable NMF problems appearing in actual applications; in such problems, $d$ is usually greater than $r$. This paper shows, without the condition $d=r$, that the preconditioned SPA is robust to noise.
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